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# Probabilities and Normal Distributions

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Part I T/F & Multiple Choice

1. The sum of all probabilities in any discrete probability distribution is not always exactly one, since some of the probabilities may be slightly larger than one. ___ T/F

2. In any binomial probability experiment, independent trials mean that the result of one trial does not affect the probability of success of any other trial in the experiment. ___ T/F

3. Under certain conditions, it is possible that the sum of the probabilities of all the sample points in a sample space is less than one. ___ T/F

4. A compound event formed by use of the word "and" requires the use of the addition rule. ____ T/F

5. If a random variable z is the standard normal score, then the standard deviation of the distribution is 1. ____ T/F

6. Every binomial distribution may be approximated reasonably by an appropriate normal distribution ___ T/F

7. If P(A) = 0.45, P(B) = 0.35, and P(A and B) = 0.25, then P(B | A) is:

A. 1.4 B. 1.8 C. 0.714 D.0.556

8. Which of the following is a characteristic of a binomial probability experiment?
A. Each trial has at least two possible outcomes
B. P(success) = 1 - P(failure)
C. The binomial random variable x is the count of the number of trials that occur
D. The result of one trial affects the probability of success on any other trial

9. If the random variable z is the standard normal score, which of the following probabilities could easily be determined without referring to a table?

A. P(z > 5)
B. P(z < 1.43)
C. P(z < - 2.95)
D. P(z > -0.35)

10. In which of the following binomial distributions is the normal approximation appropriate?

A. n = 100, p = 0.04
B. n = 50, p = 0.09
C. n = 75, p = 0.06
D. n = 12, p = 0.45

Part II. Short Answers & Computational Questions

1. Find the following probabilities:
a. Events A and B are mutually exclusive events defined on a common sample space. If P (A) = 0.3 and P(A or B) = 0.75, find P(B).
b. Events A and B are defined on a common sample space. If P(A) = 0.30, P(B) = 0.50, and P(A or B) = 0.72, find P(A and B)

2. Classify the following as discrete or continuous random variables
a. The number of Nursing majors at South University
b. The time it takes to complete this assignment
c. The blood pressures of all patients admitted to a hospital on a certain day
d. The number of people who play the state lottery each day

3. A small bag of Skittles candies has the following assortment: red (12), blue (5), orange (15), brown (0), green (16), and yellow (7). Construct the probability distribution for x.

4. Find the mean and standard deviation of the following probability distribution:

x 1 2 3
P(x) 0.4 0.15 0.35

5. In testing a new drug, researchers found that 5% of all patients using it will have a mild side effect. A random sample of 15 patients using the drug is selected. Find the probability that:
a) exactly three will have this mild side effect
b) at least two will have this mild side effect.

6. A large shipment of TV sets is accepted upon delivery if an inspection of twelve randomly selected TV sets yields no more than one defective TV.
a) Find the probability that this shipment is accepted if 5% of the total shipment is defective.
b) Find the probability that this shipment is not accepted if 10% of this shipment is defective

7. X has a normal distribution with a mean of 80.0 and a standard deviation of 3.5. Find the following probabilities:
a) P(x < 74.0)
b) P(76.0 < x < 85.0)
c) P(x>89.0)

8. Find the value of z such that 45% of the distribution lies between it and the mean

9. Assume that the average annual salary for a worker in the United States is \$31,000 and that the annual salaries for Americans are normally distributed with a standard deviation equal to \$7,500. Find the following:
a) What percentage of Americans earn below \$20,000?
b) What percentage of Americans earn above \$45,000?